Induced Interval-Valued Intuitionistic Fuzzy Hybrid Aggregation Operators with TOPSIS Order-Inducing Variables

Abstract

Two induced aggregation operators with novelly designed TOPSIS order-inducing variables are proposed: Induced Interval-valued Intuitionistic Fuzzy Hybrid Averaging (I-IIFHA) operator and Induced Interval-valued Intuitionistic Fuzzy Hybrid Geometric (I-IIFHG) operator. The merit of two aggregation operators is that they can consider additional preference information of decision maker’s attitudinal characteristics besides argument-dependent information and argument-independent information. Some desirable properties of I-IIFHA and I-IIFHG are studied and theoretical analysis also shows that they can include a wide range of aggregation operators as special cases. Further, we extend these operators to form a novel group decision-making method for selecting the most desirable alternative in multiple attribute multi-interest group decision-making problems with attribute values and decision maker’s interest values taking the form of interval-valued intuitionistic fuzzy numbers, and application research to real estate purchase selection shows its practicality.

1. Introduction

In many practical decision-making activities, the available knowledge of decision makers is vague or imprecise and cannot be assessed with numerical values, so Zadeh’s fuzzy set theory [1] and Atanassov’s intuitionistic fuzzy set (IFS) theory [2] are introduced to deal with uncertain decision environments. Over the last decades, the increasing researches have indicated that Atanassov’s IFS and interval-valued intuitionistic fuzzy set (IVIFS) [3] characterized by a membership function and nonmembership function are more suitable to deal with fuzziness and uncertainty than the generalization of fuzzy set developed by Zadeh. So recently, particular aggregation operators and corresponding decision-making methods have been studied and widely used for solving different decision-making problems under intuitionistic fuzzy or interval-valued intuitionistic fuzzy environments. As for the studies on single-person multiattribute decision-making (MADM) problems, Xu [4] investigated the methodology for solving multiattribute decision-making problems with intuitionistic fuzzy preference information aggregated by the intuitionistic fuzzy-weighted averaging (IFWA) operator, the intuitionistic fuzzy-ordered weighted averaging (IFOWA) operator, and the intuitionistic fuzzy hybrid averaging (IFHA) operator. Xu and Yager [5] proposed some geometric aggregation operators based on intuitionistic fuzzy sets, such as the intuitionistic fuzzy-weighted geometric (IFWG) operator, the intuitionistic fuzzy-ordered weighted geometric (IFOWG) operator and the intuitionistic fuzzy hybrid geometric (IFHG) operator and then applied them to multiple attribute decision-making. Xu and Xia [6] studied a method for multiattribute decision-making under IFS environment with induced generalized aggregation operators (I-GIFCOA and BS I-GIFCOA) designed by introducing Choquet integral and Dempster-Shafer evidence theory. Tan and Chen [7] considered interactions phenomena among the decision-making criteria under IFS environment and also introduced Choquet integral to design an intuitionistic fuzzy operator (I-IFC) for multicriteria decision-making, and so forth. With respect to the researches on dynamic multiattribute decision-making (DMADM) problems, Xu and Yager [8] developed two new dynamic intuitionistic fuzzy aggregation operators to solve the dynamic intuitionistic fuzzy multi-attribute decision-making problems. Wei [9] proposed the dynamic intuitionistic fuzzy-weighted geometric (DIFWG) operator and developed a procedure based on the DIFWG and IFWG operators to solve dynamic intuitionistic fuzzy multi-attribute decision-making (DIF-MADM) problems where all the decision information about attribute values takes the form of intuitionistic fuzzy numbers collected at different periods. For the researches on multiple attribute group decision-making (MAGDM) problems, Xu and Chen [10, 11] investigated decision-making methods for the IVIFS environment and defined several arithmetic aggregation operators and geometric aggregation operators, such as the interval-valued intuitionistic fuzzy-weighted averaging (IIFWA) operator, the interval-valued intuitionistic fuzzy-ordered weighted averaging (IIFOWA) operator, the interval-valued intuitionistic fuzzy hybrid averaging (IIFHA) operator, interval-valued intuitionistic fuzzy geometric (IIFG) operator, and interval-valued intuitionistic fuzzy-weighted geometric (IIFWG) operator. Wei [12] developed some induced geometric aggregation operators based on IFS and IVIFS, such as the induced intuitionistic fuzzy-ordered weighted geometric (I-IFOWG) operator and induced interval-valued intuitionistic fuzzy-ordered weighted geometric (I-IIFOWG) operator, and also extended them to solve the multi-attribute decision-making problems, and so forth.

As can be seen from the researches mentioned, aggregation operators for fusion of decision preference information play a central role in solving different kinds of decision-making problems. In the existing literatures, different types of aggregation operators and their extensions can be found for fusing intuitionistic fuzzy and interval-valued intuitionistic fuzzy information, such as the weighted aggregation operators IFWA, IFWG, IIFWA, and IIFG, the ordered weighted aggregation operators IFOWA, IFOWG, IIFOWA, and IIFWG; the induced aggregation operators I-GIFCOA, I-GIFCOA, I-IFC, I-IFOWG, and I-IIFOWG; the hybrid aggregation operators IFHA, IFHG, and IIFHA. From the aggregation process of these representative operators, it can be seen that weighted aggregation operators or ordered weighted aggregation operators can manage to utilize argument-dependent (value weights) or argument-independent (position weights) decision preference information [4, 13–17]; hybrid operators combine the advantages of both ordered weighted aggregation and weighted aggregation operators and manage to consider both position weights and value weights [17–23]; induced operators [24–32] and generalized induced operators [6, 31, 33–38] integrating order-inducing variables are able to consider additional reordering criteria that do not depend on the values of the arguments. Obviously the combination of hybrid operators and induced operators can enable the aggregation process to utilize all the argument-dependent preference information, argument-independent preference information and additional preference criteria, and so as to select the most desirable alternative in accordance with preference of decision makers. Such as, Merigó and Casanovas [20] presented an induced generalized hybrid averaging (IGHA) operator under the exact numerical environment by combination of the induced aggregation operator and the hybrid aggregation operator. But there is scarcely any research on the induced hybrid aggregation operators for the intuitionistic fuzzy or interval-valued intuitionistic fuzzy decision environment as far as can be seen from existing literatures.

In this paper, we investigate the induced hybrid operators with order-inducing variables not directly given by decision makers and their application to multiple attribute group decision-making with multi-interests under the interval-valued intuitionistic fuzzy decision environment. We present two induced interval-valued intuitionistic fuzzy hybrid aggregation operators: Induced Interval-valued Intuitionistic Fuzzy Hybrid Averaging (I-IIFHA) operator and Induced Interval-valued Intuitionistic Fuzzy Hybrid Geometric (I-IIFHG) operator. The two operators manage to use information more completely through two aspects: (1) hybridly including argument-dependent weighting vector and argument-independent weighting vector under interval-valued intuitionistic fuzzy environment, and (2) extending the IIFHA and IIFHG operators by including order-inducing variables ensured with a designed TOPSIS-based method in the aggregation process to use more attitudinal information in accordance with different decision maker’s interests.

The first main advantage of this paper is that the proposed I-IIFHA and I-IIFHG operators can help decision-making process to consider preference information in a more complete way during aggregation than other ordinary aggregation operators mentioned previously. And obviously, if the order-inducing variables are the argument ordering, then the proposed induced operators can be reduced to other ordinary aggregation operators. So the I-IIFHA and I-IIFHG operators can include a wide range of interval-valued intuitionistic fuzzy aggregation operators as special cases, such as IIFHA, IIFOWA, IIFWA, IIFHG, IIFOWG and IIFWG operators. The other important contribution of this paper is the proposal of using the information related to various decision makers’ interests to deduce order-inducing criteria in induced interval-valued intuitionistic fuzzy hybrid aggregation. Recently, although many researches have been conducted on various induced operators, there are few researches on the methods to ensure order-inducing variables. Order-inducing variables as the reordering step can enable induced aggregation operators to consider additional decision rules that do not depend on the values of the arguments, such as attitudinal characteristic information of decision makers. In the literature [27, 39, 40], Merigó and Casanovas introduced some induced aggregation operators under exact number decision environment with a complex reordering process by using Euclidean or Hamming distance measures and the man-provided order-inducing variables according to the experts’ interests. But sometimes, when decision makers cannot provide their exact preference directly because of time pressure or limited expertise related to the problem domain, certain methods should be developed to ensure order-inducing variables. Marin et al. [25] proposed to use the specificity and fuzziness measures of the unbalanced linguistic terms to induce the order-inducing variables of the arguments. For the two situations with or without man-provided additional preference, Chiclana et al. [24] presented three induced aggregation operators under exact number environment: Importance IOWA, consistency IOWA, and preference IOWA operator; the former two operators are based on the reliability of the given arguments; the latter operator is based on a relative preference associated with each one of given arguments. In this paper, we also consider the situation where the decision makers provide their own interests, but the difference is that the decision makers express their attitudinal characteristics in the form of interval-valued intuitionistic fuzzy values which can represent the information in a more complete way, and meanwhile, they can provide both ideal solutions and negative solutions. To do this, in I-IIFHA and I-IIFHG we introduce the TOPSIS method to induce the order-inducing variables, which is called TOPSIS order-inducing variables. The developed TOPSIS order-inducing method especially can also be used to induce the order-inducing variables in the situation of more than one ideal solution or negative solution provided and in the situation of no ideal solution or negative solution provided. We can use the proposed induced hybrid aggregation operators with TOPSIS order-inducing variables to utilize preference information more completely under interval-valued intuitionistic fuzzy decision environment for solving multiple attribute group decision-making problems with multi-interests.

The remainder of this paper is organized as follows. In Section 2, we give a review of basic concepts related to intuitionistic fuzzy sets and interval-valued intuitionistic fuzzy sets, and overview some aggregation operators. In Section 3, we propose induced interval-valued intuitionistic fuzzy hybrid averaging (I-IIFHA) operator and induced interval-valued intuitionistic fuzzy hybrid geometric (I-IIFHG) operator and analyze some desirable properties; then we introduce TOPSIS method to ensure order-inducing variables that reflect the complex attitudinal characteristic of the decision maker. In Section 4, based on the I-IIFHA and I-IIFHG operators, an approach is developed to solve the multiple attribute decision-making problems under the interval-valued intuitionistic fuzzy environment. In Section 5, an example is given to illustrate validity and practicality of the proposed methods. Finally, we conclude the paper.

2. Preliminaries

2.1. Related Definitions of IVIFSs

Some basic concepts on IVIFSs are introduced here in after to facilitate future discussions.

In 1986, Atanassov [2] generalized the concept of fuzzy set and defined the concept of intuitionistic fuzzy set as follows.

Definition 2.1 (see [2]). A generalized fuzzy set called intuitionistic fuzzy set is shown as follows: 𝐴=𝑥_𝑖,𝜇_𝐴𝑥_𝑖,𝜈_𝐴𝑥_𝑖∣𝑥_𝑖∈𝑋, (2.1) in which 𝜇_𝐴 means a membership function, and𝜈_𝐴 means a nonmembership, with the condition 0≤𝜇_𝐴(𝑥_𝑖)+𝜈_𝐴(𝑥_𝑖)≤1, 𝜇_𝐴(𝑥_𝑖),and𝜈_𝐴(𝑥_𝑖)∈[0,1], for all 𝑥_𝑖∈𝑋. 𝐴=𝜇_𝐴(𝑥_𝑖)=𝜈_𝐴(𝑥_𝑖) especially; the given IFS𝐴 is degraded to an ordinary fuzzy set.

In reality, it may not be easy to identify exact values for the membership and nonmembership degrees of an element to a set. In this case, a range of values may be a more appropriate measurement to accommodate the vagueness. As such, Atanassov and Gargov [3] introduce the notion of interval-valued intuitionistic fuzzy set (IVIFS).

Definition 2.2 (see [3]). An interval-valued intuitionistic fuzzy set (IVIFS) 𝐴in 𝑋 is defined by 𝐴=𝑥_𝑖,𝜇_𝐴𝑥_𝑖,𝜈_𝐴𝑥_𝑖∣𝑥_𝑖∈𝑋=𝑥_𝑖,𝜇^𝐿_𝐴𝑥_𝑖,𝜇^𝑈_𝐴𝑥_𝑖,𝜈^𝐿_𝐴𝑥_𝑖,𝜈^𝑈_𝐴𝑥_𝑖∣𝑥_𝑖∈𝑋, (2.2) where 0≤𝜇^𝐿_𝐴(𝑥_𝑖)≤𝜇^𝑈_𝐴(𝑥_𝑖)≤1,0≤𝜈^𝐿_𝐴(𝑥_𝑖)≤𝜈^𝑈_𝐴(𝑥_𝑖)≤1,0≤𝜇^𝑈_𝐴(𝑥_𝑖)+𝜈^𝑈_𝐴(𝑥_𝑖)≤1 for all 𝑥_𝑖∈𝑋. Similarity, the intervals 𝜇_𝐴(𝑥_𝑖) and 𝜈_𝐴(𝑥_𝑖) denote the degree of membership and nonmembership of an element to a set.

Then, we introduce some formulae to calculate the distance of IVIFSs.

Definition 2.3. Suppose that two interval-valued intuitionistic fuzzy sets (IVIFSs) 𝐴 and 𝐵 in 𝑋 are defined as 𝐴=𝑥_𝑖,𝜇_𝐴𝑥_𝑖,𝜈_𝐴𝑥_𝑖∣𝑥_𝑖∈𝑋=𝑥_𝑖,𝜇^𝐿_𝐴𝑥_𝑖,𝜇^𝑈_𝐴𝑥_𝑖,𝜈^𝐿_𝐴𝑥_𝑖,𝜈^𝑈_𝐴𝑥_𝑖∣𝑥_𝑖,𝑥∈𝑋𝐵=_𝑖,𝜇_𝐵𝑥_𝑖,𝜈_𝐵𝑥_𝑖∣𝑥_𝑖∈𝑋=𝑥_𝑖,𝜇^𝐿_𝐵𝑥_𝑖,𝜇^𝑈_𝐵𝑥_𝑖,𝜈^𝐿_𝐵𝑥_𝑖,𝜈^𝑈_𝐵𝑥_𝑖∣𝑥_𝑖∈𝑋. (2.3) The normalized Euclidean distance measure is 𝐷₁𝐴,𝐵=16𝑛^𝑛𝑖=1𝜇^𝐿_𝐴𝑥_𝑖−𝜇^𝐿_𝐵𝑥_𝑖²+𝜇^𝑈_𝐴𝑥_𝑖−𝜇^𝑈_𝐵𝑥_𝑖²+𝜈^𝐿_𝐴𝑥_𝑖−𝜈^𝐿_𝐵𝑥_𝑖²+ℬ+ℋ+𝒦, (2.4) where ℬ denotes(𝜈^𝑈_𝐴(𝑥_𝑖)−𝜈^𝑈_𝐵(𝑥_𝑖))², ℋ denotes(𝜋^𝐿_𝐴(𝑥_𝑖)−𝜋^𝐿_𝐵(𝑥_𝑖))², and 𝒦denotes (𝜋^𝑈_𝐴(𝑥_𝑖)−𝜋^𝑈_𝐵(𝑥_𝑖))².
The normalized Hamming distance measure is 𝐷₂𝐵=1𝐴,6𝑛^𝑛𝑖=1𝜇^𝐿_𝐴𝑥_𝑖−𝜇^𝐿_𝐵𝑥_𝑖+𝜇^𝑈_𝐴𝑥_𝑖−𝜇^𝑈_𝐵𝑥_𝑖+𝜈^𝐿_𝐴𝑥_𝑖−𝜈^𝐿_𝐵𝑥_𝑖+𝜈^𝑈_𝐴𝑥_𝑖−𝜈^𝑈_𝐵𝑥_𝑖+𝜋^𝐿_𝐴𝑥_𝑖−𝜋^𝐿_𝐵𝑥_𝑖+𝜋^𝑈_𝐴𝑥_𝑖−𝜋^𝑈_𝐵𝑥_𝑖. (2.5) The normalized Hausdorff distance measure is 𝐷₃𝐵=1𝐴,𝑛^𝑛𝑖=1max𝜇^𝐿_𝐴𝑥_𝑖−𝜇^𝐿_𝐵𝑥_𝑖,𝜇^𝑈_𝐴𝑥_𝑖−𝜇^𝑈_𝐵𝑥_𝑖,𝜈^𝐿_𝐴𝑥_𝑖−𝜈^𝐿_𝐵𝑥_𝑖,𝜈^𝑈_𝐴𝑥_𝑖−𝜈^𝑈_𝐵𝑥_𝑖,𝜋^𝐿_𝐴𝑥_𝑖−𝜋^𝐿_𝐵𝑥_𝑖,𝜋^𝑈_𝐴𝑥_𝑖−𝜋^𝑈_𝐵𝑥_𝑖. (2.6)

In order to aggregate interval-valued intuitionistic fuzzy information, we introduce the following relations and operations given by Xu [10].

Definition 2.4 (see [10]). Let 𝛼=([𝑎,𝑏],[𝑐,𝑑]), 𝛼₁=([𝑎₁,𝑏₁],[𝑐₁,𝑑₁]), 𝛼₂=([𝑎₂,𝑏₂],[𝑐₂,𝑑₂]) be three IVIFNs; then(1)𝛼₁⊕𝛼₂=([𝑎₁+𝑎₂−𝑎₁𝑎₂,𝑏₁+𝑏₂−𝑏₁𝑏₂],[𝑐₁𝑐₂,𝑑₁𝑑₂]);(2)𝛼₁⊗𝛼₂=([𝑎₁𝑎₂,𝑏₁𝑏₂],[𝑐₁+𝑐₂−𝑐₁𝑐₂,𝑑₁+𝑑₂−𝑑₁𝑑₂]);(3)𝜆𝛼=([1−(1−𝑎)^𝜆,1−(1−𝑏)^𝜆],[𝑐^𝜆,𝑑^𝜆]);(4)𝛼^𝜆=([𝑎^𝜆,𝑏^𝜆],[1−(1−𝑐)^𝜆,1−(1−𝑑)^𝜆]).

Definition 2.5 (see [10]). Let 𝛼=([𝑎,𝑏],[𝑐,𝑑]), 𝛼₁=([𝑎₁,𝑏₁],[𝑐₁,𝑑₁]), 𝛼₂=([𝑎₂,𝑏₂],[𝑐₂,𝑑₂]) be three IVIFNs, 𝜆,𝜆₁,𝜆₂≥0; then(1)𝛼₁⊕𝛼₂=𝛼₂⊕𝛼₁;(2)𝛼₁⊗𝛼₂=𝛼₂⊗𝛼₁;(3)𝜆(𝛼₁⊕𝛼₂)=𝜆𝛼₂⊕𝜆𝛼₁;(4)(𝛼₁⊗𝛼₂)^𝜆=𝛼₂^𝜆⊗𝛼₁^𝜆;(5)𝜆₁𝛼⊕𝜆₂𝛼=(𝜆₁+𝜆₂)𝛼;(6)𝛼^𝜆1⊗𝛼^𝜆2=(𝛼)^𝜆1+𝜆2.

In order to rank alternatives, it is necessary to consider how to compare two IVIFNs. Xu [10] also devised an approach to compare two IVIFNs based on the concepts of score function and accuracy function.

Definition 2.6 (see [10]). For any three IVIFNs 𝛼=([𝜇^𝐿,𝜇^𝑈],[𝜈^𝐿,𝜈^𝑈]), 𝛼₁=([𝜇^𝐿₁,𝜇^𝑈₁],[𝜈^𝐿₁,𝜈^𝑈₁]), 𝛼₂=([𝜇^𝐿₂,𝜇^𝑈₂],[𝜈^𝐿₂,𝜈^𝑈₂]), score function defined as 𝑠(𝛼)=(1/2)(𝜇^𝐿+𝜇^𝑈−𝜈^𝐿−𝜈^𝑈), and accuracy function defined as ℎ(𝛼)=(1/2)(𝜇^𝐿+𝜇^𝑈+𝜈^𝐿+𝜈^𝑈).If 𝑠(𝛼₁)<𝑠(𝛼₂), then 𝛼₁ is smaller than 𝛼₂, denoted by 𝛼₁<𝛼₂.If 𝑠(𝛼₁)=𝑠(𝛼₂), then one has the following. If ℎ(𝛼₁)<ℎ(𝛼₂), then 𝛼₁ is smaller than 𝛼₂, denoted by 𝛼₁<𝛼₂. If ℎ(𝛼₁)=ℎ(𝛼₂), then 𝛼₁ and 𝛼₂ represent the same information, denoted by 𝛼₁=𝛼₂.

2.2. Some Interval-Valued Intuitionistic Fuzzy Aggregation Operators

We first introduce definitions of interval-valued intuitionistic fuzzy-weighted averaging (IIFWA) and interval-valued intuitionistic fuzzy-weighted geometric (IIFWG) operators.

Definition 2.7 (see [10]). Let 𝛼₁,𝛼₂,…,𝛼_𝑛 be a collection of interval arguments, where 𝛼_𝑗=([𝑎_𝑗,𝑏_𝑗],[𝑐_𝑗,𝑑_𝑗]), and IIFWA operator of dimension 𝑛 is a mapping IIFWA: 𝑅^𝑛→𝑅, that has an associated vector 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛)^𝑇 such that ∑^𝑛_𝑗=1𝑤_𝑗=1 and 𝑤_𝑗∈[0,1]; then IIFWA_𝜔𝛼₁,𝛼₂,…,𝛼_𝑛=𝜔₁𝛼₁⊕𝜔₂𝛼₂⊕⋯⊕𝜔_𝑛𝛼_𝑛=1−^𝑛_𝑗=11−𝑎_𝑗^𝜔𝑗,1−^𝑛_𝑗=11−𝑏_𝑗^𝜔𝑗,^𝑛_𝑗=1𝑐_𝑗^𝜔𝑗,^𝑛_𝑗=1𝑑_𝑗^𝜔𝑗. (2.7)

Definition 2.8 (see [10]). Let 𝛼₁,𝛼₂,…,𝛼_𝑛 be a collection of interval arguments, where 𝛼_𝑗=([𝑎_𝑗,𝑏_𝑗],[𝑐_𝑗,𝑑_𝑗]), and IIFWG operator of dimension 𝑛 is a mapping IIFWG: 𝑅^𝑛→𝑅, that has an associated weight vector 𝜔=(𝜔₁,𝜔₂,…,𝜔_𝑛)^𝑇 such that ∑^𝑛_𝑗=1𝜔_𝑗=1 and 𝜔_𝑗∈[0,1]; then IIFWG_𝜔𝛼₁,𝛼₂,…,𝛼_𝑛=𝛼₁^𝜔1⊗𝛼₂^𝜔2⊗⋯⊗𝛼_𝑛^𝜔𝑛=^𝑛_𝑗=1𝑎_𝑗^𝜔𝑗,^𝑛_𝑗=1𝑏_𝑗^𝜔𝑗,1−^𝑛_𝑗=11−𝑐_𝑗^𝜔𝑗,1−^𝑛_𝑗=11−𝑑_𝑗^𝜔𝑗. (2.8)

Furthermore, consider the ordered positions of the interval-valued intuitionistic fuzzy values instead of weighting the interval-valued intuitionistic fuzzy values themselves. Xu and Chen [11] also proposed interval-valued intuitionistic fuzzy-ordered weighted averaging (IIFOWA) and interval-valued intuitionistic fuzzy-ordered weighted geometric (IIFOWG) operators.

Definition 2.9 (see [11]). Let 𝛼₁,𝛼₂,…,𝛼_𝑛 be a collection of interval arguments, where 𝛼_𝑗=([𝑎_𝑗,𝑏_𝑗],[𝑐_𝑗,𝑑_𝑗]), and IIFOWA operator of dimension 𝑛 is a mapping IIFOWA: 𝑅^𝑛→𝑅, that has an associated weight vector 𝜔=(𝜔₁,𝜔₂,…,𝜔_𝑛)^𝑇 such that ∑^𝑛_𝑗=1𝜔_𝑗=1 and 𝜔_𝑗∈[0,1]; then IIFOWA_𝜔𝛼₁,𝛼₂,…,𝛼_𝑛=𝜔₁𝛼_𝜋(1)⊕𝜔₂𝛼_𝜋(2)⊕⋯⊕𝜔_𝑛𝛼_𝜋(𝑛)=1−^𝑛_𝑗=11−𝑎_𝜋(𝑗)^𝜔𝑗,1−^𝑛_𝑗=11−𝑏_𝜋(𝑗)^𝜔𝑗,^𝑛_𝑗=1𝑐^𝜔_𝜋(𝑗)^𝑗,^𝑛_𝑗=1𝑑^𝜔_𝜋(𝑗)^𝑗, (2.9) where 𝛼_𝜋(1),𝛼_𝜋(2),…,𝛼_𝜋(𝑛) is a permutation of 𝛼₁,𝛼₂,…,𝛼_𝑛, such that 𝛼_{𝜋(𝑗−1)}≥𝛼_𝜋(𝑗) for all 𝑗.

Definition 2.10 (see [11]). Let 𝛼₁,𝛼₂,…,𝛼_𝑛 be a collection of interval arguments, where 𝛼_𝑗=([𝑎_𝑗,𝑏_𝑗],[𝑐_𝑗,𝑑_𝑗]), and IIFOWG operator of dimension 𝑛 is a mapping IIFOWG: 𝑅^𝑛→𝑅, that has an associated weight vector 𝜔=(𝜔₁,𝜔₂,…,𝜔_𝑛)^𝑇 such that ∑^𝑛_𝑗=1𝜔_𝑗=1 and 𝜔_𝑗∈[0,1]; then IIFOWG_𝜔𝛼₁,𝛼₂,…,𝛼_𝑛=𝛼^𝜔_𝜋(1)¹⊗𝛼^𝜔_𝜋(2)²⊗⋯⊗𝛼^𝜔_𝜋(𝑛)^𝑛=^𝑛_𝑗=1𝑎^𝜔_𝜋(𝑗)^𝑗,^𝑛_𝑗=1𝑏^𝜔_𝜋(𝑗)^𝑗,1−^𝑛_𝑗=11−𝑐_𝜋(𝑗)^𝜔𝑗,1−^𝑛_𝑗=11−𝑑_𝜋(𝑗)^𝜔𝑗, (2.10) where 𝛼_𝜋(1),𝛼_𝜋(2),…,𝛼_𝜋(𝑛) is a permutation of 𝛼₁,𝛼₂,…,𝛼_𝑛, such that 𝛼_{𝜋(𝑗−1)}≥𝛼_𝜋(𝑗) for all 𝑗.

Ordered weighted aggregation operators only consider the position weights of the interval-valued intuitionistic fuzzy values themselves. To overcome this limitation, Xu and Chen [11] provided definitions of interval-valued intuitionistic fuzzy hybrid averaging (IIFHA) operator and interval-valued intuitionistic fuzzy hybrid geometric (IIFHG) operator, which weights both the given interval-valued intuitionistic fuzzy value and its ordered position information.

Definition 2.11 (see [11]). An IIFHA operator of dimension 𝑛 is a mapping IIFHA: 𝑅^𝑛→𝑅 that has an associated vector 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛)^𝑇 such that ∑^𝑛_𝑗=1𝑤_𝑗=1 and 𝑤_𝑗∈[0,1]; then IIFHA_𝜔,𝑤𝛼₁,𝛼₂,…,𝛼_𝑛=𝑤₁𝛼̇_𝜋(1)⊕𝑤₂𝛼̇_𝜋(2)⊕⋯⊕𝑤_𝑛𝛼̇_𝜋(𝑛), (2.11) where 𝛼̇_𝜋(𝑗) is the 𝑗th largest of the weighted intuitionistic fuzzy values 𝛼̇_𝑗=𝑛𝜔_𝑗𝛼_𝑗, 𝜔=(𝜔₁,𝜔₂,…,𝜔_𝑛)^𝑇 is the weight vector 𝛼_𝑗(𝑗=1,2,…,𝑛) with ∑^𝑛_𝑗=1𝜔_𝑗=1 and 𝜔_𝑗∈[0,1], and 𝑛 is the balancing coefficient.

Definition 2.12 (see [11]). An IIFHG operator of dimension 𝑛 is a mapping IIFHG: 𝑅^𝑛→𝑅 that has an associated vector 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛)^𝑇 such that ∑^𝑛_𝑗=1𝑤_𝑗=1 and 𝑤_𝑗∈[0,1]; then IIFHG_𝜔,𝑤𝛼₁,𝛼₂,…,𝛼_𝑛=𝛼̇^𝑤_𝜋(1)¹⊗𝛼̇^𝑤_𝜋(2)²⊗⋯⊗𝛼̇^𝑤_𝜋(𝑛)^𝑛, (2.12) where 𝛼̇_𝜋(𝑗) is the 𝑗th largest of the weighted intuitionistic fuzzy values 𝛼̇_𝑗=𝛼_𝑗^𝑛𝜔𝑗, 𝜔=(𝜔₁,𝜔₂,…,𝜔_𝑛)^𝑇 is the weight vector 𝛼_𝑗(𝑗=1,2,…,𝑛) with ∑^𝑛_𝑗=1𝜔_𝑗=1 and 𝜔_𝑗∈[0,1], and 𝑛 is the balancing coefficient.

The induced operator is an extension of the ordinary aggregation operators. The main difference is that the reordering step is not carried out with the arguments values. In this case, the reordering step is developed with order-inducing variables that reflect a more complex reordering process. Wei and Yi [29] introduced the induced interval-valued intuitionistic fuzzy-ordered weighted geometric (I-IIFOWG) operator, which is defined as follows.

Definition 2.13 (see [29]). An I-IIFOWG operator of dimension 𝑛 is a mapping I-IIFOWG: 𝑅^𝑛→𝑅 that has an associated vector 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛)^𝑇 such that ∑^𝑛_𝑗=1𝑤_𝑗=1 and 𝑤_𝑗∈[0,1]; then I-IIFOWG_𝑤𝜀₁,𝛼₁,𝜀₂,𝛼₂,…,𝜀_𝑛,𝛼_𝑛=𝛼^𝑤_𝜋(1)¹⊕𝛼^𝑤_𝜋(2)²⊕⋯⊕𝛼^𝑤_𝜋(𝑛)^𝑛=^𝑛_𝑗=1𝑎^𝑤_𝜋(𝑗)^𝑗,^𝑛_𝑗=1𝑏^𝑤_𝜋(𝑗)^𝑗,1−^𝑛_𝑗=11−𝑐_𝜋(𝑗)^𝑤𝑗,1−^𝑛_𝑗=11−𝑑_𝜋(𝑗)^𝑤𝑗, (2.13) where 𝛼_𝜋(𝑗)=([𝑎_𝜋(𝑗),𝑏_𝜋(𝑗)],[𝑐_𝜋(𝑗),𝑑_𝜋(𝑗)]) is the 𝛼 value of the I-IIFOWG pair ⟨𝜀_𝑖,𝛼_𝑖⟩ having the 𝑗th largest 𝜀_𝑖(𝜀_𝑖∈[0,1]), and 𝜀_𝑖 in ⟨𝜀_𝑖,𝛼_𝑖⟩ is referred to as the order-inducing variable and 𝛼_𝑖=([𝑎_𝑖,𝑏_𝑖],[𝑐_𝑖,𝑑_𝑖]) as the interval-valued intuitionistic fuzzy values.

In the same way, we have the definition of interval-valued intuitionistic fuzzy-ordered weighted averaging (I-IIFOWA) operator.

Definition 2.14. An I-IIFOWA operator of dimension 𝑛 is a mapping I-IIFOWA: 𝑅^𝑛→𝑅, that has an associated vector with I-IIFOWA operator 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛)^𝑇 such that ∑^𝑛_𝑗=1𝑤_𝑗=1 and 𝑤_𝑗∈[0,1]; then I-IIFOWA_𝑤𝜀₁,𝛼₁,𝜀₂,𝛼₂,…,𝜀_𝑛,𝛼_𝑛=𝑤₁𝛼_𝜋(1)⊕𝑤₂𝛼_𝜋(2)⊕⋯⊕𝑤_𝑛𝛼_𝜋(𝑛)=1−^𝑛_𝑗=11−𝑎_𝜋(𝑗)^𝑤𝑗,1−^𝑛_𝑗=11−𝑏_𝜋(𝑗)^𝑤𝑗,^𝑛_𝑗=1𝑐^𝑤_𝜋(𝑗)^𝑗,^𝑛_𝑗=1𝑑^𝑤_𝜋(𝑗)^𝑗, (2.14) where 𝛼_𝜋(𝑗)=([𝑎_𝜋(𝑗),𝑏_𝜋(𝑗)],[𝑐_𝜋(𝑗),𝑑_𝜋(𝑗)]) is the 𝛼 value of the I-IIFOWA pair ⟨𝜀_𝑖,𝛼_𝑖⟩ having the 𝑗th largest 𝜀_𝑖(𝜀_𝑖∈[0,1]), and 𝜀_𝑖 in ⟨𝜀_𝑖,𝛼_𝑖⟩ is referred to as the order-inducing variable and 𝛼_𝑖=([𝑎_𝑖,𝑏_𝑖],[𝑐_𝑖,𝑑_𝑖])as the interval-valued intuitionistic fuzzy values.

Xu [5] classified all these approaches for determining the weighting vector in aggregation operators into two categories: argument-independent weighting vector method and argument-dependent weighting vector method. The weights derived by the argument-independent approaches are associated with particular ordered positions of the aggregated arguments and have no connection with the aggregated arguments, while the argument-dependent approaches determine the weights based on the arguments.

In this paper, the argument-weighting vector 𝜔=(𝜔₁,𝜔₂,…,𝜔_𝑛)^𝑇 associated with the inputs 𝛼_𝑗 can be determined by the normal distribution method [13].

Let 𝛼₁,𝛼₂,…,𝛼_𝑛 be a collection of interval arguments, where 𝛼_𝑗=([𝑎_𝑗,𝑏_𝑗],[𝑐_𝑗,𝑑_𝑗]), the mean of these interval-valued intuitionistic fuzzy sets 𝑢=([𝑎_𝑢,𝑏_𝑢],[𝑐_𝑢,𝑑_𝑢]), where𝑎_𝑢=1𝑛^𝑛_𝑗=1𝑎_𝑗,𝑏_𝑢=1𝑛^𝑛_𝑗=1𝑏_𝑗,𝑐_𝑢=1𝑛^𝑛_𝑗=1𝑐_𝑗,𝑑_𝑢=1𝑛^𝑛_𝑗=1𝑑_𝑗, (2.15) and the variance of these interval-valued intuitionistic fuzzy arguments:𝜎=1𝑛^𝑛𝑗=1𝑑𝛼_𝑗,𝑢². (2.16) then we can have a normal distribution method for deriving the attribute weight associated with its value:𝜔_𝑗=1√𝑒2𝜋𝜎^−𝑑2(𝛼_𝑗−𝑢_)/2𝜎²,𝑗=1,2,…,𝑛. (2.17) Consider that 𝜔_𝑗∈[0,1] and ∑^𝑛_𝑗=1𝜔_𝑗=1; we have normalized attribute weights:𝜔_𝑗=1/√2𝜋𝜎𝑒^−𝑑2(𝛽_𝑗−𝑢_)/2𝜎²∑^𝑛_𝑗=11/√2𝜋𝜎𝑒^−𝑑2(𝛽_𝑗−𝑢_)/2𝜎²,𝑗=1,2,…,𝑛. (2.18)

The position-weighting vector 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛)^𝑇 associated with the operator can be determined by RIM quantifiers [41]:𝑄(𝑟)=𝑟^𝛼𝑤,𝛼≥0,_𝑖𝑖=𝑄𝑛−𝑄𝑖−1𝑛,for𝑖=1,2,…,𝑛. (2.19)

3. Induced Interval-Valued Intuitionistic Fuzzy Hybrid Aggregation Operators with TOPSIS Order-Inducing Variables

3.1. I-IIFHA and I-IIFHG Operators

Induced interval-valued intuitionistic fuzzy hybrid operators are the extensions of ordinary interval-valued intuitionistic fuzzy hybrid operators, and the difference resides in that they can consider more preference information during aggregation process by combination of order-inducing variables independent of arguments. Here, based on the IIFHA and IIFHG, we propose two more general formulations of the reordering process to form I-IIFHA and I-IIFHG operators, which can be defined as follows.

Definition 3.1. An I-IIFHA operator of dimension 𝑛 is a mapping I-IIFHA: 𝑅^𝑛→𝑅 defined by an associated weighting vector 𝑤 of dimension 𝑛 such that the sum of the weights is 1 and 𝑤_𝑗∈[0,1], a set of order-inducing variables 𝜀_𝑗, according to the following formula: I-IIFHA_𝜔,𝑤𝜀₁,𝛼₁,𝜀₂,𝛼₂,…,𝜀_𝑛,𝛼_𝑛=𝑤₁𝛼̇_𝜋(1)⊕𝑤₂𝛼̇_𝜋(2)⊕⋯⊕𝑤_𝑛𝛼̇_𝜋(𝑛)=1−^𝑛_𝑗=11−𝑎̇_𝜋(𝑗)^𝑤𝑗,1−^𝑛_𝑗=11−𝑏̇̃_𝜋(𝑗)^𝑤𝑗,^𝑛_𝑗=1𝑐̇̃^𝑤_𝜋(𝑗)^𝑗,^𝑛_𝑗=1𝑑̇^𝑤_{𝜋(𝑗)𝑗}, (3.1) where 𝛼̇_𝑖=([𝛼̇_𝑖,𝑏̇̃_𝑖],[𝑐_𝑖̇̃,𝑑̇_𝑖])=𝑛𝜔_𝑖𝛼_𝑖=([1−(1−𝑎_𝑖)^𝑛𝜔𝑖,1−(1−𝑏̃_𝑖)^𝑛𝜔𝑖],[𝑐̃_𝑖^𝑛𝜔𝑖,𝑑_𝑖^𝑛𝜔𝑖]),(𝛼̇_𝜋(1),𝛼̇_𝜋(2),…,𝛼̇_𝜋(𝑛)) is (𝛼₁,𝛼₂,…,𝛼_𝑛) reordered in decreasing order of the order-inducing variables values of the 𝜀_𝑗, 𝜔=(𝜔₁,𝜔₂,…,𝜔_𝑛)^𝑇 is the weight vector 𝛼_𝑗(𝑗=1,2,…,𝑛) with ∑^𝑛_𝑗=1𝜔_𝑗=1 and 𝜔_𝑗∈[0,1], and 𝑛 is the balancing coefficient.

Definition 3.2. An I-IIFHG operator of dimension 𝑛 is a mapping I-IIFHG: 𝑅^𝑛→𝑅 defined by an associated weighting vector 𝑤 of dimension 𝑛 such that the sum of the weights is 1 and 𝑤_𝑗∈[0,1], a set of order-inducing variables 𝜀_𝑗, according to the following formula: I-IIFHG_𝜔,𝑤𝜀₁,𝛼₁,𝜀₂,𝛼₂,…,𝜀_𝑛,𝛼_𝑛=^𝑛_𝑗=1𝛼̇^𝑤_𝜋(𝑗)^𝑗=^𝑛_𝑗=1𝛼̇^𝑤_𝜋(𝑗)^𝑗,^𝑛_𝑗=1𝑏̇̃^𝑤_𝜋(𝑗)^𝑗,1−^𝑛_𝑗=11−𝑐̇̃_𝜋(𝑗)^𝑤𝑗,1−^𝑛_𝑗=11−𝑑̇_𝜋(𝑗)^𝑤𝑗, (3.2) where 𝛼̇_𝑖=([𝛼̇_𝑖,𝑏̇̃_𝑖],[𝑐_𝑖̇̃,𝑑̇_𝑖])=𝛼_𝑖^𝑛𝜔𝑖=([𝑎_𝑖^𝑛𝜔𝑖,𝑏_𝑖^𝑛𝜔𝑖],[1−(1−𝑐_𝑖)^𝑛𝜔𝑖,1−(1−𝑑_𝑖)^𝑛𝜔𝑖]), (𝛼̇_𝜋(1),𝛼̇_𝜋(2),…,𝛼̇_𝜋(𝑛)) is (𝛼₁,𝛼₂,…,𝛼_𝑛) reordered in decreasing order of the order-inducing variables values of the 𝜀_𝑗, 𝜔=(𝜔₁,𝜔₂,…,𝜔_𝑛)^𝑇 is the weight vector 𝛼_𝑖(𝑖=1,2,…,𝑛) with ∑^𝑛_𝑖=1𝜔_𝑖=1 and 𝜔_𝑖∈[0,1], and 𝑛 is the balancing coefficient.

Remark 3.3. The argument-weighting vector 𝜔=(𝜔₁,𝜔₂,…,𝜔_𝑛)^𝑇 associated with inputs (𝛼₁,𝛼₂,…,𝛼_𝑛) can be determined by normal distribution method defined in (2.17) and (2.18); The position-weighting vector 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛)^𝑇 associated with the operator can be determined by RIM quantifiers in (2.19).

3.2. TOPSIS Order-Inducing Variables

Order-inducing variables as reordering step can enable induced aggregation operators to consider additional decision rules that do not depend on the values of the arguments. In this subsection, we propose a TOPSIS-based method to ensure order-inducing variables to be included in I-IIFHA and I-IIFHG operators; as such we can be able to deal with the complex decision-making problems in which the solution with highest score value is not the optimal one for the decision makers.

For example, we have the temperature problem introduced by Marin et al. [25] under linguistic environment. If we are dealing with temperatures of the human body, the optimal term is adequate; if the temperature is higher, the results are not good; and on the other side, if the temperature is lower, the results are also not good. If we are dealing with temperatures at which we want to travel, different persons may be interested in a different ordering of the temperatures; a person may prefer to visit places with cold weather, whereas another person may prefer to warm weather. So proper order-inducing variables should be introduced to deal with the problems where the solution with highest values may not be the optimal one, because of additional decision criteria to influence the ordering of arguments, such as rules of nature and preference of person.

In the induced operators mentioned previously, such as IOWA operator, ILOWA operator, and I-IIFOWA operator, the order-inducing variables have been researched under different environments of exact numbers, interval-valued numbers, or linguistic numbers. Here, we let the decision maker define his own preference by interval-valued intuitionistic fuzzy numbers; then a TOPSIS-based method is proposed as follows to derive the order-inducing variables, which is called TOPSIS order-inducing variables.

Step 1. Let the decision maker define the interval-valued intuitionistic fuzzy ideal solution(s) and the interval-valued intuitionistic fuzzy negative ideal solution(s) corresponding to a collection of interval-valued intuitionistic interval arguments (𝛼₁,𝛼₂,…,𝛼_𝑛): 𝛼⁺_𝑗=𝛼⁺_𝑗1,𝛼⁺_𝑗2,…,𝛼⁺_𝑗𝑝,𝛼⁻_𝑗=𝛼⁻_𝑗1,𝛼⁻_𝑗2,…,𝛼⁻_𝑗𝑞, (3.3) where 𝑝(𝑝≥0) is the number of ideal solution(s), and 𝑞(𝑞≥0) is the number of negative solution(s).

Step 2. Calculate the distance(s) between the interval-valued intuitionistic interval arguments (𝛼₁,𝛼₂,…,𝛼_𝑛) and ideal solution(s) and negative solution(s): 𝑑⁺_𝑗=max_{𝑙=1,2,…,𝑝}𝑑𝛼_𝑗,𝛼⁺_𝑗𝑙,𝑑⁻_𝑗=min_{𝑙=1,2,…,𝑞}𝑑𝛼_𝑗,𝛼⁻_𝑗𝑞, (3.4) where 𝑑(𝛼_𝑗,𝛼⁺_𝑗𝑙) and 𝑑(𝛼_𝑗,𝛼⁻_𝑗𝑙) can be calculated by the distance measures defined in Definition 2.3. If there is more than one ideal solution, we choose the maximum distance one(s); if there exist more than one negative solution, we choose the minimum distance one(s).

Step 3. Calculate the coefficients 𝑐_𝑗=𝑑⁻_𝑗𝑑⁻_𝑗+𝑑⁺_𝑗,𝑗=1,2,…,𝑛. (3.5)

Step 4. Derive the order-inducing variable 𝑢_𝑗 by descending order of 𝑐_𝑗. The greater the value 𝑐_𝑗, the bigger the ordering-induced variable 𝑢_𝑗.

Remark 3.4. In decision-making problems, one decision maker has no interval-valued intuitionistic fuzzy ideal solution(s) or negative solution(s), and we assume that 𝑑⁺_𝑗=1 and 𝑑⁻_𝑗=0 if the attribute is the benefit type and 𝑑⁺_𝑗=0 and 𝑑⁻_𝑗=1 if it is the cost type.

3.3. Properties

Let 𝑓^𝐴 be the I-IIFHA operator, and let 𝑓^𝐺 be the I-IIFHG operator; 𝜔=(𝜔₁,𝜔₂,…,𝜔_𝑛)^𝑇 is argument-weighting vector, and 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛)^𝑇is position-weighting vector. One has two interval-valued intuitionistic fuzzy vectors:𝐴=𝜀_𝛼(1),𝛼₁,𝜀_𝛼(2),𝛼₂,…,𝜀_𝛼(𝑛),𝛼_𝑛,𝐵=𝜀_𝛽(1),𝛽₁,𝜀_𝛽(2),𝛽₂,…,𝜀_𝛽(𝑛),𝛽_𝑛. (3.6) Then we can get these theorems of I-IIFHA and I-IIFHG operator as follows.

Theorem 3.5 (monotonicity). The I-IIFHA and I-IIFHG operators are increasingly monotonous with respect to the argument values if the associated order-inducing variables remain unchanged. 𝑓^𝐴_𝑤,𝜔𝜀_𝛼(1),𝛼₁,𝜀_𝛼(2),𝛼₂,…,𝜀_𝛼(𝑛),𝛼_𝑛≥𝑓^𝐴_𝑤,𝜔𝜀_𝛽(1),𝛽₁,𝜀_𝛽(2),𝛽₂,…,𝜀_𝛽(𝑛),𝛽_𝑛,𝑓^𝐺_𝑤,𝜔𝜀_𝛼(1),𝛼₁,𝜀_𝛼(2),𝛼₂,…,𝜀_𝛼(𝑛),𝛼_𝑛≥𝑓^𝐺_𝑤,𝜔𝜀_𝛽(1),𝛽₁,𝜀_𝛽(2),𝛽₂,…,𝜀_𝛽(𝑛),𝛽_𝑛,𝑖𝑓𝛼_𝑗≥𝛽_𝑗,𝜀_𝛼(𝑗)=𝜀_𝛽(𝑗)∀𝑗. (3.7)

Proof. Thus, one has 𝛼_𝑗≥𝛽_𝑗⇒𝛼̇_𝑗=𝛼_𝑗^{𝑛𝜔𝛼(𝑗)}≥𝛽̇_𝑗=𝛽_𝑗^{𝑛𝜔𝛽(𝑗)}. And 𝜀_𝛼(𝑗)=𝜀_𝛽(𝑗), 𝑓^𝐴_𝑤,𝜔𝜀_𝛼(1),𝛼₁,𝜀_𝛼(2),𝛼₂,…,𝜀_𝛼(𝑛),𝛼_𝑛≥𝑓^𝐴_𝑤,𝜔𝜀_𝛽(1),𝛽₁,𝜀_𝛽(2),𝛽₂,…,𝜀_𝛽(𝑛),𝛽_𝑛,𝑓^𝐺_𝑤,𝜔𝜀_𝛼(1),𝛼₁,𝜀_𝛼(2),𝛼₂,…,𝜀_𝛼(𝑛),𝛼_𝑛≥𝑓^𝐺_𝑤,𝜔𝜀_𝛽(1),𝛽₁,𝜀_𝛽(2),𝛽₂,…,𝜀_𝛽(𝑛),𝛽_𝑛. (3.8)

Theorem 3.6 (commutativity). The I-IIFHA and I-IIFHG operators are commutative. 𝑓^𝐴_𝑤,𝜔𝜀_𝛼(1),𝛼₁,𝜀_𝛼(2),𝛼₂,…,𝜀_𝛼(𝑛),𝛼_𝑛=𝑓^𝐴_𝑤,𝜔𝜀_𝛽(1),𝛽₁,𝜀_𝛽(2),𝛽₂,…,𝜀_𝛽(𝑛),𝛽_𝑛,𝑓^𝐺_𝑤,𝜔𝜀_𝛼(1),𝛼₁,𝜀_𝛼(2),𝛼₂,…,𝜀_𝛼(𝑛),𝛼_𝑛=𝑓^𝐺_𝑤,𝜔𝜀_𝛽(1),𝛽₁,𝜀_𝛽(2),𝛽₂,…,𝜀_𝛽(𝑛),𝛽_𝑛, (3.9) where (𝛼₁,𝛼₂,…,𝛼_𝑛) is any permutation of the arguments (𝛽₁,𝛽₂,…,𝛽_𝑛).

Proof. Since (𝛼₁,𝛼₂,…,𝛼_𝑛) is a permutation of (𝛽₁,𝛽₂,…,𝛽_𝑛), then we have 𝛼̇_𝑗=𝑛𝜔_𝛼(𝑗)𝛼_𝑗=𝛽̇_𝑗=𝑛𝜔_𝛽(𝑗)𝛽_𝑗 and 𝛼̇_𝑗=𝛼_𝑗^{𝑛𝜔𝛼(𝑗)}=𝛽̇_𝑗=𝛽_𝑗^{𝑛𝜔𝛽(𝑗)} for all 𝑗. Meanwhile, (𝜀_𝛼(1),𝜀_𝛼(2),…,𝜀_𝛼(𝑛)) is equal to (𝜀_𝛽(1),𝜀_𝛽(2),…,𝜀_𝛽(𝑛)) which is associated with the decision maker’s preference, so 𝑤_𝛼(1)𝛼̇₁⊕𝑤_𝛼(2)𝛼̇₂⊕⋯⊕𝑤_𝛼(𝑛)𝛼̇_𝑛=𝑤_𝛽(1)𝛽̇₁⊕𝑤_𝛽(2)𝛽̇₂⊕⋯⊕𝑤_𝛽(𝑛)𝛽̇_𝑛,𝛼̇₁^𝑤𝛼(1)⊗𝛼̇₂^𝑤𝛼(2)⊗⋯⊗𝛼̇_𝑛^{𝑤𝛼(𝑛)}=𝛽̇₁^𝑤_𝛽(1)⊗𝛽̇₂^𝑤_𝛽(2)⊗⋯⊗𝛽̇_𝑛^𝑤_𝛽(𝑛). (3.10) We get 𝑓^𝐴_𝑤,𝜔𝜀_𝛼(1),𝛼₁,𝜀_𝛼(2),𝛼₂,…,𝜀_𝛼(𝑛),𝛼_𝑛=𝑓^𝐴_𝑤,𝜔𝜀_𝛽(1),𝛽₁,𝜀_𝛽(2),𝛽₂,…,𝜀_𝛽(𝑛),𝛽_𝑛,𝑓^𝐺_𝑤,𝜔𝜀_𝛼(1),𝛼₁,𝜀_𝛼(2),𝛼₂,…,𝜀_𝛼(𝑛),𝛼_𝑛=𝑓^𝐺_𝑤,𝜔𝜀_𝛽(1),𝛽₁,𝜀_𝛽(2),𝛽₂,…,𝜀_𝛽(𝑛),𝛽_𝑛. (3.11)

Theorem 3.7 (idempotency). The I-IIFHA and I-IIFHG operators are idempotent. If 𝛼_𝑗=𝛼 for all 𝑗, then 𝑓^𝐴_𝑤,𝜔𝜀_𝛼(1),𝛼₁,𝜀_𝛼(2),𝛼₂,…,𝜀_𝛼(𝑛),𝛼_𝑛𝑓=𝛼,_{𝐺𝑤,𝜔}𝜀_𝛼(1),𝛼₁,𝜀_𝛼(2),𝛼₂,…,𝜀_𝛼(𝑛),𝛼_𝑛=𝛼. (3.12)

Proof. Since 𝛼_𝑗=𝛼 for all 𝑗, we have (𝜔₁,𝜔₂,…,𝜔_𝑛)=(1/𝑛,1/𝑛,…,1/𝑛), which is associated with the input arguments (𝛼₁,𝛼₂,…,𝛼_𝑛); then 𝑓^𝐴_𝑤,𝜔𝜀_𝛼(1),𝑛𝜔₁𝛼₁,𝜀_𝛼(2),𝑛𝜔₂𝛼₂,…,𝜀_𝛼(𝑛),𝑛𝜔_𝑛𝛼_𝑛=𝑓^𝐴_𝑤,𝜔𝜀_𝛼(1),𝛼₁,𝜀_𝛼(2),𝛼₂,…,𝜀_𝛼(𝑛),𝛼_𝑛=𝑤₁𝛼_𝜋(1)⊕𝑤₂𝛼_𝜋(2)⊕⋯⊕𝑤_𝑛𝛼_𝜋(𝑛)=𝑤₁+𝑤₂+⋯+𝑤_𝑛𝛼=𝛼,𝑓^𝐺_𝑤,𝜔𝜀_𝛼(1),𝛼₁^𝑛𝜔1,𝜀_𝛼(2),𝛼₂^𝑛𝜔2,…,𝜀_𝛼(𝑛),𝛼_𝑛^𝑛𝜔𝑛=𝑓^𝐺_𝑤,𝜔𝜀_𝛼(1),𝛼₁,𝜀_𝛼(2),𝛼₂,…,𝜀_𝛼(𝑛),𝛼_𝑛=𝛼^𝑤_𝜋(1)¹⊗𝛼^𝑤_𝜋(2)²⊗⋯⊗𝛼^𝑤_𝜋(𝑛)^𝑛=𝛼^{(𝑤1+𝑤2+⋯+𝑤𝑛)}=𝛼. (3.13)

Theorem 3.8 (bounded). The I-IIFHA and I-IIFHG operators are bounded. That is, for any weighting vector 𝜔 and 𝑤: min𝛼₁,𝛼₂,…,𝛼_𝑛≤𝑓^𝐴_𝑤,𝜔𝜀_𝛼(1),𝛼₁,𝜀_𝛼(2),𝛼₂,…,𝜀_𝛼(𝑛),𝛼_𝑛≤max𝛼₁,𝛼₂,…,𝛼_𝑛,min𝛼₁,𝛼₂,…,𝛼_𝑛≤𝑓^𝐺_𝑤,𝜔𝜀_𝛼(1),𝛼₁,𝜀_𝛼(2),𝛼₂,…,𝜀_𝛼(𝑛),𝛼_𝑛≤max𝛼₁,𝛼₂,…,𝛼_𝑛. (3.14)

Proof. By definitions of I-IIFHA and I-IIFHG operators and operations of IVIFNs, we can obtain that min𝛼_𝑖≤min𝛼_𝜋(𝑖)≤𝑓^𝐴_𝑤,𝜔𝜀_𝛼(1),𝛼₁,𝜀_𝛼(2),𝛼₂,…,𝜀_𝛼(𝑛),𝛼_𝑛≤max𝛼_𝜋(𝑖)≤max𝛼_𝑖,min𝛼_𝑖≤min𝛼_𝜋(𝑖)≤𝑓^𝐺_𝑤,𝜔𝜀_𝛼(1),𝛼₁,𝜀_𝛼(2),𝛼₂,…,𝜀_𝛼(𝑛),𝛼_𝑛≤max𝛼_𝜋(𝑖)≤max𝛼_𝑖. (3.15)

Theorem 3.9. If 𝜔=(1/𝑛,1/𝑛,…,1/𝑛), I-IIFHA and I-IIFHG operators reduce to I-IIFOWA and I-IIFOWG operators. If 𝑤=(1/𝑛,1/𝑛,…,1/𝑛), they reduce to IIFWA and IIFWG operators. If there was no any positive or negative preference, (𝜀_𝛼(1),𝜀_𝛼(2),…,𝜀_𝛼(𝑛))=(𝜀_𝛽(1),𝜀_𝛽(2),…,𝜀_𝛽(𝑛)), the IIFHA and IIFHG operators are the special cases of I-IIFHA and I-IIFHG operators.

Proof. Let 𝜔=(1/𝑛,1/𝑛,…,1/𝑛); then I-IIFHA𝜀_𝛼(1),𝑛𝜔₁𝛼₁,𝜀_𝛼(2),𝑛𝜔₂𝛼₂,…,𝜀_𝛼(𝑛),𝑛𝜔_𝑛𝛼_𝑛=𝑤₁𝛼_𝜋(1)⊕𝑤₂𝛼_𝜋(2)⋯⊕𝑤_𝑛𝛼_𝜋(𝑛)=I−IIFOWA_𝑤𝛼₁,𝛼₂,…,𝛼_𝑛,I-IIFHG𝜀_𝛼(1),𝛼₁^𝑛𝜔1,𝜀_𝛼(2),𝛼₂^𝑛𝜔2,…,𝜀_𝛼(𝑛),𝛼_𝑛^𝑛𝜔𝑛=𝛼^𝑤_𝜋(1)¹⊗𝛼^𝑤_𝜋(2)²⊗⋯⊗𝛼^𝑤_𝜋(𝑛)^𝑛=I−IIFOWG_𝑤𝛼₁,𝛼₂,…,𝛼_𝑛. (3.16) Let 𝑤=(1/𝑛,1/𝑛,…,1/𝑛); then I-IIFHA𝜀_𝛼(1),𝑛𝜔₁𝛼₁,𝜀_𝛼(2),𝑛𝜔₂𝛼₂,…,𝜀_𝛼(𝑛),𝑛𝜔_𝑛𝛼_𝑛=𝜔₁𝛼₁⊕𝜔₂𝛼₂⊕⋯⊕𝜔_𝑛𝛼_𝑛=IIFWA_𝜔𝛼₁,𝛼₂,…,𝛼_𝑛,I-IIFHG𝜀_𝛼(1),𝛼₁^𝑛𝜔1,𝜀_𝛼(2),𝛼₂^𝑛𝜔2,…,𝜀_𝛼(𝑛),𝛼_𝑛^𝑛𝜔𝑛=𝛼₁^𝜔1⊗𝛼₂^𝜔2⊗⋯⊗𝛼_𝑛^𝜔𝑛=IIFWG_𝜔𝛼₁,𝛼₂,…,𝛼_𝑛. (3.17)

Theorem 3.10. In the situation of 𝜈^𝐿_𝑗=0 or 𝜈^𝐿_𝑗=0, interval-valued intuitionistic fuzzy geometric operators are better than interval-valued intuitionistic fuzzy averaging operators, while in 𝜇^𝐿_𝑗=0 or 𝜇^𝑈_𝑗=0, interval-valued intuitionistic fuzzy averaging operators are better than interval-valued intuitionistic fuzzy geometric operators.

Proof. Take the IIFWA and IIFWG operators, for example.
If 𝜈^𝐿_𝑗=0 and 𝜈^𝐿_𝑗=0,IIFWA_𝜔𝛼₁,𝛼₂,…,𝛼_𝑛=𝜔₁𝛼₁⊕𝜔₂𝛼₂⊕⋯⊕𝜔_𝑛𝛼_𝑛=1−^𝑛_𝑗=11−𝜇^𝐿_𝑗^𝜔𝑗,1−^𝑛_𝑗=11−𝜇^𝑈_𝑗^𝜔𝑗,^𝑛_𝑗=1𝜈^𝐿_𝑗^𝜔𝑗,^𝑛_𝑗=1𝜈^𝑈_𝑗^𝜔𝑗=1−^𝑛_𝑗=11−𝜇^𝐿_𝑗^𝜔𝑗,1−^𝑛_𝑗=11−𝜇^𝑈_𝑗^𝜔𝑗,[0,0]. (3.18) If 𝜇^𝐿_𝑗=0 and 𝜇^𝑈_𝑗=0, IIFWG_𝜔𝛼₁,𝛼₂,…,𝛼_𝑛=𝛼₁^𝜔1⊗𝛼₂^𝜔2⊗⋯⊗𝛼_𝑛^𝜔𝑛=^𝑛_𝑗=1𝜇^𝐿_𝑗^𝜔𝑗,^𝑛_𝑗=1𝜇^𝑈_𝑗^𝜔𝑗,1−^𝑛_𝑗=11−𝜈^𝐿_𝑗^𝜔𝑗,1−^𝑛_𝑗=11−𝜈^𝑈_𝑗^𝜔𝑗=[0,0],1−^𝑛_𝑗=11−𝜈^𝐿_𝑗^𝜔𝑗,1−^𝑛_𝑗=11−𝜈^𝑈_𝑗^𝜔𝑗. (3.19) Similarly, this rule is also suitable for IIFOWA, IIFOWG, IIFHA, IIFHG, I-IIFOWA, I-IIFOWG, I-IIFHA, I-IIFHG operators, and so forth.

4. Application in Interval-Valued Intuitionistic Fuzzy Multiple-Attribute Group Decision-Making

For a multiple-attribute group decision-making problem, let 𝑋={𝑥₁,𝑥₂,…,𝑥_𝑛} be a set of alternatives, and let 𝐺={𝑔₁,𝑔₂,…,𝑔_𝑚} be a set of criteria. Let 𝐷={𝑑₁,𝑑₂,…,𝑑_𝑡} be a set of decision makers, and let 𝜆=(𝜆⁽¹⁾,𝜆⁽²⁾,…,𝜆^(𝑡))^𝑇 be the weighting vector of decision makers. In the following, we apply the I-IIFHA and I-IIFHG operators (whose position-weighting vector is𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑚)^𝑇 to multiple-attribute group decision-making based on interval-valued intuitionistic fuzzy information. The approach involves the following steps.

Step 1. Construct the interval-valued intuitionistic fuzzy evaluation matrix.

𝑅=(𝑟̃_𝑖𝑗)_𝑛×𝑚=(𝜇_𝑖𝑗,𝜈_𝑖𝑗)_𝑛×𝑚=([𝜇^𝐿_𝑖𝑗,𝜇^𝑈_𝑖𝑗],[𝜈^𝐿_𝑖𝑗,𝜈^𝑈_𝑖𝑗])_𝑛×𝑚, where [𝜇^𝐿_𝑖𝑗,𝜇^𝑈_𝑖𝑗] indicates the degree at wich the alternative 𝑥_𝑖 satisfies the attribute 𝑔_𝑗 given by the decision maker, and [𝜈^𝐿_𝑖𝑗,𝜈^𝑈_𝑖𝑗] indicates the degree at which the alternative 𝑥_𝑖(𝑖=1,2,…,𝑛) does not satisfy the attribute 𝑔_𝑗(𝑗=1,2,…,𝑚).

Step 2. Obtain the interval-valued intuitionistic fuzzy positive-ideal vector and negative-ideal vector corresponding to each alternative’s attribute by every decision maker.
Let the 𝑘th decision maker provide the interval-valued intuitionistic fuzzy positive-ideal vector 𝑅^(𝑘)+ and the interval-valued intuitionistic fuzzy negative-ideal vector 𝑅^(𝑘)−𝑅^(𝑘)+=𝑟̃₁^(𝑘)+,𝑟̃₂^(𝑘)+,…,𝑟̃_𝑗^(𝑘)+,…,𝑟̃_𝑚^(𝑘)+,𝑅^(𝑘)−=𝑟̃₁^(𝑘)−,𝑟̃₂^(𝑘)−,…,𝑟̃_𝑗^(𝑘)−,…,𝑟̃_𝑚^(𝑘)−, (4.1) where 𝑟̃_𝑗^(𝑘)+=𝑟̃^(𝑘)+_𝑗1,𝑟̃^(𝑘)+_𝑗2,…,𝑟̃^(𝑘)+_𝑗𝑙,…,𝑟̃^(𝑘)+_𝑗𝑝,𝑟̃_𝑗^(𝑘)−=𝑟̃^(𝑘)−_𝑗1,𝑟̃^(𝑘)−_𝑗2,…,𝑟̃^(𝑘)−_𝑗𝑙,…,𝑟̃^(𝑘)−_𝑗𝑞. (4.2)

Step 3. Calculate the argument-weighting vector 𝜔=(𝜔₁,𝜔₂,…,𝜔_𝑛)^𝑇 associated with the interval-valued intuitionistic fuzzy value 𝑟̃_𝑖𝑗 by normal distribution method.

Step 4. Construct aggregated weighted interval-valued intuitionistic fuzzy decision matrix 𝑅̇=(𝑟̇̃_𝑖𝑗)_𝑛×𝑚: ̇𝑔𝑅=¹𝑔₂⋯𝑔_𝑚𝑥₁𝑟̇̃₁₁𝑟̇̃₁₂⋯𝑟̇̃_1𝑚𝑥₂𝑟̇̃₂₁𝑟̇̃₂₂⋯𝑟̇̃_2𝑚𝑥⋮⋮⋮⋮⋮_𝑛𝑟̇̃_𝑛1𝑟̇̃_𝑛2⋯𝑟̇̃_𝑛𝑚=𝑟̇̃_{𝑖𝑗𝑛×𝑚}, (4.3) where 𝑟̇̃_𝑖𝑗=𝑛𝜔_𝑗𝑟̃_𝑖𝑗 or 𝑟̇̃_𝑖𝑗=𝑟̃_𝑖𝑗^𝑛𝜔𝑗, and 𝜔=(𝜔₁,𝜔₂,…,𝜔_𝑛)^𝑇 is the weighting vector associated with interval-valued intuitionistic fuzzy value 𝑟̃_𝑖𝑗 itself.

Step 5. Ensure the TOPSIS order-inducing variables 𝜀 by using of positive-ideal and negative-ideal information provided by each decision maker in Step 2.
First, we calculate the distance between the evaluation information 𝑟̃_𝑖𝑗 and 𝑘th decision maker’s preference information. 𝑑^(𝑘)+_𝑖𝑗=max_{𝑙=1,2,…,𝑝}𝑑𝑟̃_𝑖𝑗,𝑟̃^(𝑘)+_𝑗𝑙,𝑑^(𝑘)−_𝑖𝑗=min_{𝑙=1,2,…,𝑞}𝑑𝑟̃_𝑖𝑗,𝑟̃^(𝑘)−_𝑗𝑙 (4.4) where 𝑑(𝑟̃_𝑖𝑗,𝑟̃_𝑗^(𝑘)+) and 𝑑(𝑟̃_𝑖𝑗,𝑟̃_𝑗^(𝑘)−) can be calculated by the normalized Euclidean distance measure in Definition 2.3.
We can calculate the coefficients of each alternative’s attribute: 𝑐^(𝑘)_𝑖𝑗=𝑑^(𝑘)−_𝑖𝑗𝑑^(𝑘)−_𝑖𝑗+𝑑^(𝑘)+_𝑖𝑗,𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚,𝑘=1,2,…,𝑡, (4.5) and we can get coefficients matrix 𝐶^(𝑘)=(𝑐^(𝑘)_𝑖𝑗)_𝑛×𝑚. By ascending order of 𝑐^(𝑘)_𝑖𝑗, we get the ordering induced variables matrix 𝜀^(𝑘)=(𝜀^(𝑘)_𝑖𝑗)_𝑛×𝑚.

Step 6. Utilize the aggregated weighted interval-valued intuitionistic fuzzy decision matrix 𝑅̇=(𝑟̇̃_𝑖𝑗)_𝑛×𝑚 and I-IIFHA or I-IIFHG operator to derive the 𝑘th decision maker overall interval-valued intuitionistic fuzzy arguments 𝑅^(𝑘)=(𝑟̃₁^(𝑘),𝑟̃₂^(𝑘),…,𝑟̃_𝑛^(𝑘)) corresponding to each alternative 𝑥_𝑖: 𝑟̃_𝑖^(𝑘)=𝜇_𝑖^(𝑘)𝐿,𝜇_𝑖^(𝑘)𝑈,𝜈_𝑖^(𝑘)𝐿,𝜈_𝑖^(𝑘)𝑈=I-IIFHA_𝜔,𝑤𝜀^(𝑘)_𝑖1,𝑟̃^(𝑘)_𝑖1,𝜀^(𝑘)_𝑖2,𝑟̃^(𝑘)_𝑖2,…,𝜀^(𝑘)_𝑖𝑚,𝑟̃^(𝑘)_𝑖𝑚or𝑟̃_𝑖^(𝑘)=𝜇_𝑖^(𝑘)𝐿,𝜇_𝑖^(𝑘)𝑈,𝜈_𝑖^(𝑘)𝐿,𝜈_𝑖^(𝑘)𝑈=I-IIFHG_𝜔,𝑤𝜀^(𝑘)_𝑖1,𝑟̃^(𝑘)_𝑖1,𝜀^(𝑘)_𝑖2,𝑟̃^(𝑘)_𝑖2,…,𝜀^(𝑘)_𝑖𝑚,𝑟̃^(𝑘)_𝑖𝑚. (4.6)

Step 7. Utilize the individual overall interval-valued intuitionistic fuzzy arguments vector 𝑟̃_𝑖^(𝑘) and IIFWA or IIFWG operator to derive the group overall interval-valued intuitionistic fuzzy argument vector 𝑟_𝑖̃: 𝑟_𝑖̃=𝜇^𝐿_𝑖,𝜇^𝑈_𝑖,𝜈^𝐿_𝑖,𝜈^𝑈_𝑖=IIFWA𝑟̃_𝑖⁽¹⁾,𝑟̃_𝑖⁽²⁾,…,𝑟̃_𝑖^(𝑡)or𝑟_𝑖̃=𝜇^𝐿_𝑖,𝜇^𝑈_𝑖,𝜈^𝐿_𝑖,𝜈^𝑈_𝑖=IIFWG𝑟̃_𝑖⁽¹⁾,𝑟̃_𝑖⁽²⁾,…,𝑟̃_𝑖^(𝑡). (4.7)

Step 8. Calculate the scores 𝑠(𝑟_𝑖̃)(𝑖=1,2,…,𝑛) and the accuracy values ℎ(𝑟_𝑖̃)(𝑖=1,2,…,𝑛) of the alternative 𝑥_𝑖(𝑖=1,2,…,𝑛), respectively.

Step 9. Utilize the scores 𝑠(𝑟_𝑖̃)(𝑖=1,2,…,𝑛) and the accuracy values ℎ(𝑟_𝑖̃)(𝑖=1,2,…,𝑛) to rank all the alternatives 𝑥_𝑖(𝑖=1,2,…,𝑛) and then to select the optimal one(s).

5. Applied Example

Here we take the real estate purchase for applied example. There are five candidate houses denoted by 𝑥_𝑖(𝑖=1,2,3,4,5) that can be evaluated by the IVIFNs 𝑟̃_𝑖𝑗 with respect to the five attributes: (1)𝑔₁: size; (2)𝑔₂: price; (3)𝑔₃: structure; (4)𝑔₄: position; and (5)𝑔₅: environment, as listed in Table 1. In the real situation, the DMs may have his/her preference to each attribute. For example, someone may prefer the house close to suburb, while someone may prefer the position of urban area. DMs give their interval-valued intuitionistic fuzzy ideal values and negative ideal values corresponding to five attributes shown in Tables 2, 3, and 4. Suppose that the expert-weighting vector is 𝜆=(0.3,0.3,0.4)^𝑇.

Table 1 

Decision matrix 𝑅 with IVIFNs.

Table 2 

Preference decision matrix 𝑅⁽¹⁾ provided by 𝑑₁.

Table 3 

Preference decision matrix 𝑅⁽²⁾ provided by 𝑑₂.

Table 4 

Preference decision matrix 𝑅⁽³⁾ provided by 𝑑₃.

Then we can get the alternative’s attribute coefficients matrix 𝐶⁽¹⁾, 𝐶⁽²⁾, and 𝐶⁽³⁾ as shown in Tables 5, 6, and 7 according to (4.4) and (4.5).

Table 5 

Alternative’s attribute coefficient matrix 𝐶⁽¹⁾.

Table 6 

Alternative’s attribute coefficient matrix 𝐶⁽²⁾.

Table 7 

Alternative’s attribute coefficient matrix 𝐶⁽³⁾.

By the normal distribution method, we can determine argument-weighting vector associated with their own values:𝜔=(0.2225,0.1911,0.2421,0.1516,0.1927)^𝑇. (5.1)

There exist 𝜈^𝐿_𝑥𝑖𝑗=0 and 𝜈^𝑈_𝑥𝑖𝑗=0 in interval-valued intuitionistic fuzzy decision matrix 𝑅; we choose I-IIFHG and IIFWG operators to aggregate each decision information according to Theorem 3.10.

We first get the weighted interval-valued intuitionistic fuzzy decision matrix 𝑅̇=(𝑟̇̃_𝑖𝑗)_𝑛×𝑚 as shown in Table 8.

Table 8 

Aggregate results 𝑅̇.

Step 6. Suppose the position-weighting vector 𝑤=(0.4472,0.1853,0.1421,0.1198,0.1056)^𝑇 generalized by RIM quantifier. Utilize the aggregated weighted interval-valued intuitionistic fuzzy decision matrix 𝑅̇=(𝑟̇̃_𝑖𝑗)_𝑛×𝑚 and I-IIFHG operator to derive the 𝑘th decision maker overall interval-valued intuitionistic fuzzy arguments 𝑅^(𝑘)=(𝑟̃₁^(𝑘),𝑟̃₂^(𝑘),…,𝑟̃_𝑛^(𝑘)) corresponding to each alternative 𝑥_𝑖(𝑖=1,2,…,5). 𝑟̃₁⁽¹⁾=([0.5041,0.6482],[0.1449,0.2952]),𝑟̃₂⁽¹⁾=([0.3410,0.5936],[0.1862,0.3348]),𝑟̃₃⁽¹⁾=([0.4483,0.6866],[0.0704,0.1991]),𝑟̃₄⁽¹⁾=([0.3709,0.5783],[0.1253,0.2871]),𝑟̃₅⁽¹⁾=([0.4741,0.6772],[0.0806,0.2151]);𝑟̃₁⁽²⁾=([0.4583,0.6614],[0.1668,0.3184]),𝑟̃₂⁽²⁾=([0.3311,0.6091],[0.1840,0.3358]),𝑟̃₃⁽²⁾=([0.4246,0.6685],[0.1071,0.2202]),𝑟̃₄⁽²⁾=([0.4439,0.6469],[0.0898,0.1914]),𝑟̃₅⁽²⁾=([0.4741,0.6772],[0.0806,0.2151]);𝑟̃₁⁽³⁾=([0.4920,0.6994],[0.1344,0.2729]),𝑟̃₂⁽³⁾=([0.3311,0.6091],[0.1840,0.3358]),𝑟̃₃⁽³⁾=([0.4483,0.6866],[0.0704,0.1991]),𝑟̃₄⁽³⁾=([0.3858,0.5923],[0.1091,0.2616]),𝑟̃₅⁽³⁾=([0.4633,0.6618],[0.1073,0.2351]). (5.2)

Step 7. Use the individual overall interval-valued intuitionistic fuzzy arguments vector 𝑟̃_𝑖^(𝑘) and IIFWG operator to derive the group overall interval-valued intuitionistic fuzzy arguments vector 𝑟_𝑖̃(𝑖=1,2,…,5): 𝑟₁̃=([0.4829,0.6635],[0.1517,0.3002]),𝑟₂̃=([0.3350,0.6029],[0.1849,0.3354]),𝑟₃̃=([0.4387,0.6793],[0.0853,0.2076]),𝑟₄̃=([0.4017,0.6077],[0.1080,0.2450]),𝑟₅̃=([0.4719,0.6741],[0.0860,0.2191]). (5.3)

Step 8. Calculate the scores 𝑠(𝑟_𝑖̃)(𝑖=1,2,…,5)of the collective overall intuitionistic fuzzy values by Definition 2.6. 𝑠𝑟₁̃=0.6945,𝑠𝑟₂̃=0.4176,𝑠𝑟₃̃=0.8251,𝑠𝑟₄̃=0.6564,𝑠𝑟₅̃=0.8409. (5.4)

Step 9. Rank all the alternatives𝑥_𝑖(𝑖=1,2,…,5) in accordance with the scores 𝑠(𝑟_𝑖̃)(𝑖=1,2,…,5) of the collective overall intuitionistic fuzzy values: 𝑥₂≺𝑥₄≺𝑥₁≺𝑥₃≺𝑥₅, and thus the most desirable alternative is 𝑥₅.
In the previous analysis, the computation of illustrative example is conducted according to the proposed hybrid induced I-IIFHG operator. And we have also conducted computation according to other representative operators: two induced operators of I-IIFOWA and I-IIFOWG, which can consider the attitudinal characteristics of experts and the argument-independent preference information; two hybrid operators of IIFHA and IIFHG, which can consider the argument-dependent and argument-independent preference information; and the other proposed induced hybrid I-IIFHA operator. Comparative aggregated results and orderings are listed in Tables 9 and 10.

Table 9 

Aggregated results by different operators.

Table 10 

Ordering of the candidate houses.

As can be seen from the comparative results in Table 10, because the last four hybrid operators (IIFHA, IIFHG, I-IIFHA, and I-IIFHG) manage to give more consideration to the argument-dependent preference information than the first two induced operators (I-IIFOWA and I-IIFOWG), the 5th candidate alternative is emphasized to be the optimal one. And among the last four hybrid operators, because the last two induced hybrid operators (I-IIFHA and I-IIFHG) manage to give more consideration to the additional attitudinal characteristics of experts than the two hybrid operators (IIFHA and IIFHG), the 3rd alternative happens to meet the experts’ interests and is emphasized to be better than 1st, 2nd, and 4th alternatives. Also it is can be seen from Table 10, the optimal alternative is consistent with all the last four operators and the worst alternative is consistent with all the six comparative operators.

Generally different aggregation operators for different decision environments may obtain different orderings of candidate solutions. The I-IIFHA and I-IIFHG operators can hybridly aggregate the argument-dependent preference information, argument-independent preference information, and the additional attitudinal characteristics of experts, and can be reduced to different ordinary operators under different decision environments. So the main advantage of the proposed I-IIFHA and I-IIFHG operator with TOPSIS order-inducing variables is that they can include a wide range of particular cases and so can consider a lot of possibilities with a more complete view of the different scenarios when solving multiple-attribute multi-interest group decision-making problems under interval-valued intuitionistic fuzzy decision environment.

6. Conclusion

We have presented a decision-making approach based on the proposed I-IIFHA and I-IIFHG operators. I-IIFHA and I-IIFHG operators are extensions of the IIFHA and IIFHG operators by using the TOPSIS order-inducing variables in the reordering process. The prominent main advantage of these operators is that they are able to consider more complex attitudinal characteristics in the decision process than other ordinary operators. We have studied some desirable properties of the I-IIFHA and I-IIFHG operators, such as commutativity, monotonicity, idempotency, and bounded. Theoretical analysis has shown that I-IIFHA and I-IIFHG can include a wide range of aggregation operators as special cases, such as IIFHA, IIFOWA, IIFWA, IIFHG, IIFOWG, and IIFWG operators. Furthermore, we have applied these operators to form an approach for the multiple attribute multi-interest group decision-making problems with interval-valued intuitionistic fuzzy information. In the future, we will computerize the proposed methods to facilitate the usage and continue working on the extension and application of developed operators to different environments and domains.

Acknowledgments

This work was supported by the Social Science Foundation of Ministry of Education of China under Grant 11YJC630283 and Zhejiang Provincial Natural Science Foundation of China under Grant Y6110345.